This result, after dividing by fifteen to reduce it to seconds of time, must be added to the half-interval between the times of observation in the morning and in the afternoon, when the distance of the celestial body from the elevated pole has increased, but subtracted when that distance has diminished.

As to the declinations themselves. First, for the sun, the declination is given in the Nautical Almanac both for apparent noon and for mean noon, with differences for one hour; the difference for any interval can then be applied to obtain the required reduced declination for any corresponding moment. The time used in the Nautical Almanac being Greenwich time, the local time of observation must first be reduced to the corresponding Greenwich time by applying the longitude expressed in time; the difference of declination for the interval will be either added to or subtracted from the noon declination used according as the declination is increasing or decreasing, or as the Greenwich date precedes or follows the nearest noon. Secondly, for the moon, whose declination is given for every hour, and for the planets, whose declination is given for every day at mean noon, the reduction is effected in the same way. Thirdly, the declinations of certain stars are given for January 1st, with their annual variations, to which the fraction of the year can be applied as for right ascensions, as before explained.

The method of equal altitudes is highly accurate, and is independent of previous observations.

A more common and coarser method of finding the time, assumes that the latitude is known, and is based on a single altitude. The observation is made on a celestial body of known declination, when on or near the prime vertical, that is, nearly due east or due west.

of the observer, as under those circumstances the apparent motion in altitude is greatest, and thus gives most precision in result.

Two or three such altitudes' taken with a sextant or a reflecting repeating circle, either from the sea horizon or from an artificial mercury horizon, are corrected for index error, horizon, refraction, and parallax, and their mean obtained; the complement of this is then the co-altitude, or zenith distance, forming a side of the spherical triangle under consideration; the corresponding times of observation and their mean are also noted.

Now the quantity required is the interval in time. between the azimuthal direction of the celestial body and the meridian of the observer, which is represented in angular measurement by the angle ZPS in the attached figure 40, where Z is the zenith, P the elevated pole, S the sun or star.

The three sides of the triangle ZPS are given, for ZS is the zenith distance or co altitude, ZP is the co

1 For explanation of the process of taking and correcting altitudes, see the following paragraph on Latitude.

latitude, and PS the distance of the celestial body from

the elevated pole is equal to 90+ its declination.

Then if the required hour-angle ZPS=h,

The obtained hour-angle is then reduced to its equivalent in time, which is the interval between the time of observation and the time of apparent noon, whether earlier or later as the case may be, according as the celestial body was rising or falling at the time of observation. The difference one way or the other between twenty-four hours and the interval in time is the apparent local time of the observation, and if this be corrected by applying the equation of time, the result is the true local time of the observation. This, compared with the recorded time shown by the chronometer, shows the chronometer error at the moment of mean observation.

This last method differs from the former in that it utilises the altitudes observed, instead of neutralising them; they hence require careful correction. This subject is explained in the following paragraph on Latitude.

These methods of finding the time are illustrated by examples following the explanation of this branch of the subject.

In addition to the above, the following is a convenient and ready mode of checking from time to time the rate of a watch or chronometer that is known to go fairly. Having chosen some easily found familiar star (and a second one as a reserve when required) observe

and note its altitude when near the prime vertical by sextant or reflecting circle and artificial horizon; and note the time by the watch, supposed in the first instance to be corrected by results of previous observations.

If the rate of watch-error is to be found every three, seven, ten, or fourteen days, set the angular instrument to the angle noted on the required day, and observing the same star, note the time of passage by the watch. For acceleration of the star in mean time subtract 3 minutes 55908 seconds for every day in the interval between the two observations from the time noted at the first observation; the difference will be the correct time for the second observation. Compare this calculated time with the observed time to obtain the error gaining or losing in the interval, and thence deduce the daily rate of gain or loss for the watch.

The Latitude,

The simplest mode of obtaining the latitude is by observing the meridian altitude of any celestial body whose declination is given in the Nautical Almanac ; the sum of, or the difference between, the meridian altitude and the declination, will give either the latitude or the complement of the latitude.

The following is a rule that avoids hesitation among the four cases. Subtract the corrected altitude from 90°, and thus obtain the true zenith distance, which is then marked as north or south, according as the zenith is north or south of the celestial body; then, if the declination is similarly marked in the Nautical Almanac, the sum of the two is the latitude, marked in the same way, but if the declination is differently marked, the

difference will be the latitude, and will be marked either north or south like the greater of the two arcs employed.

This method of obtaining the latitude is the most common one; it depends on a single, or on a mean of several, closely following, observations of altitude, which hence must be fully corrected to obtain a true altitude.

The principles on which these corrections are based require examination in detail.

In the first place the instrumental correction for index error, if any, must be applied, although the sextant or circle may be in good adjustment. The resulting angle should then be divided by two if an artificial horizon has been used; but if the sea horizon has been used a correction for dip must be supplied, which may be obtained from tables, or calculated. The eye of the observer being at a certain height (/) above the true horizon, the horizon is hence depressed in proportion to that height, but this is also modified by terrestrial refraction, which renders the correction uncertain in amount, and hence to be avoided whenever possible. Its value is nearly

when is the radius of the earth, and it is subtractive.

The remaining corrections for altitudes to be applied afterwards are three in number; first for refraction, second for parallax, and third for semi-diameter.

Refraction is due to the effect of the atmosphere in bending a ray of light, thus forcing us to see a celestial object at an apparent elevation which is higher than its true position; hence the correction for refraction must be subtracted from the observed altitude. Its amount varies with the tangent of the zenith distance of the body observed, but also depends on the humidity,